Abstract

We consider the application of tomography to the reconstruction of 2-D vector fields. The most convenient sensor configuration in such problems is the regular positioning along the domain boundary. However, the most accurate reconstructions are obtained by sampling uniformly the Radon parameter domain rather than the border of the reconstruction domain. This dictates a prohibitively large number of sensors and impractical sensor positioning. In this paper, we propose uniform placement of the sensors along the boundary of the reconstruction domain and interpolation of the measurements for the positions that correspond to uniform sampling in the Radon domain. We demonstrate that when the cubic spline interpolation method is used, a 60 times reduction in the number of sensors may be achieved with only about 10% increase in the error with which the vector field is estimated. The reconstruction error by using the same sensors and ignoring the necessity of uniform sampling in the Radon domain is in fact higher by about 30%. The effects of noise are also examined.

© 2010 Optical Society of America

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  1. S. P. Juhlin, “Doppler tomography,” Proceedings of the 15th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, October 28–31, 1993, San Diego, California, USA (1993) pp. 212–213.
    [CrossRef]
  2. Y. K. Tao, A. M. Davis, and J. A. Izatt, “Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified Hilbert transform,” Opt. Express 16,12350–12361(2008).
    [CrossRef] [PubMed]
  3. B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: Mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
    [CrossRef]
  4. W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Philos. Trans. R. Soc. London, Ser. A 307, 439–464 (1982).
    [CrossRef]
  5. D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833(1991).
    [CrossRef]
  6. S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.
  7. D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” ISA Trans. 26, 2674–2677(1979).
    [CrossRef]
  8. S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201–233(1982).
    [CrossRef] [PubMed]
  9. S. J. Norton, “Tomographic reconstruction of 2-D vector fields: Application to flow imaging,” Geophys. J. Int. 97, 161–168 (1988).
    [CrossRef]
  10. S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
    [CrossRef] [PubMed]
  11. H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471(1991).
    [CrossRef]
  12. K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38(1990).
    [CrossRef]
  13. K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33(1993).
    [CrossRef] [PubMed]
  14. M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246(1994).
    [CrossRef]
  15. H. M. Hertz, “Kerr effect tomography for nonintrusive spatially resolved measurements of asymmetric electric field distributions,” Appl. Opt. 25, 914–921(1986).
    [CrossRef] [PubMed]
  16. H. K. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924(1987).
    [CrossRef] [PubMed]
  17. N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transf. 53, 723–728(1995).
    [CrossRef]
  18. G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212(1988).
    [CrossRef] [PubMed]
  19. V. A. Sharafutdinov, “Tomographic problem of photoelasticity,” Proc. SPIE 1843, 234–243(1992).
    [CrossRef]
  20. H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221(1997).
    [CrossRef]
  21. A. Schwarz, “Three-dimensional reconstruction of temperature and velocity fields in a furnace,” Proceedings of ECAPT, The European Concerted Action on Process Tomography (International Society for Industrial Process Tomography, 1994), pp. 227–233.
  22. S. E. Segre, “The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave,” Plasma Phys. 20, 295–307(1978).
    [CrossRef]
  23. P. Juhlin, “Principles of Doppler Tomography,” Lund Institute of Technology, Sweden, Department of Mathematics, LUTFD2/(TFMA-92)/7002+17P, August (1992).
  24. M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-d vector fields using a system of linear equations,” Proceedings of the 12th Annual Medical Image Understanding and Analysis Conference (MIUA 2008) July 2-3, 2008, Dundee, Scotland, UK, pp. 132–136.
  25. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).
  26. M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 30–44(2004).
    [CrossRef] [PubMed]
  27. A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1631–1645(2006).
    [CrossRef] [PubMed]
  28. D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing(American Mathematical Society, 2002).
  29. H. R. Schwarz, Numerische Mathematik (B. G. Teubner, 1986).
  30. H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).
  31. I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).
  32. C. de Boor, A Practical Guide to Splines (Springer-Verlag, 1978).
    [CrossRef]
  33. C. Habermann and F. Kindermann, “Multidimensional spline interpolation: Theory and applications,” Comput. Econ. 30, 153–169(2007).
    [CrossRef]
  34. R. M. Rangayyan, M. Ciuc, and F. Faghih, “Adapted-neighborhood filtering of images corrupted by signal-dependent noise,” Appl. Opt. 37, 4477–4487(1998).
    [CrossRef]
  35. T. D. Sanger, J. Kaiser, and B. Placek, “Reaching movements in childhood dystonia contain signal-dependent noise,” J. Child Neurol. 20, 489–496(2005).
    [PubMed]
  36. G. Krüger and G. H. Glover, “Physiological noise in oxygenation-sensitive magnetic resonance imaging,” Magn. Reson. Med. 46, 631–677(2001).
    [CrossRef] [PubMed]

2008 (2)

M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-d vector fields using a system of linear equations,” Proceedings of the 12th Annual Medical Image Understanding and Analysis Conference (MIUA 2008) July 2-3, 2008, Dundee, Scotland, UK, pp. 132–136.

Y. K. Tao, A. M. Davis, and J. A. Izatt, “Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified Hilbert transform,” Opt. Express 16,12350–12361(2008).
[CrossRef] [PubMed]

2007 (1)

C. Habermann and F. Kindermann, “Multidimensional spline interpolation: Theory and applications,” Comput. Econ. 30, 153–169(2007).
[CrossRef]

2006 (1)

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1631–1645(2006).
[CrossRef] [PubMed]

2005 (1)

T. D. Sanger, J. Kaiser, and B. Placek, “Reaching movements in childhood dystonia contain signal-dependent noise,” J. Child Neurol. 20, 489–496(2005).
[PubMed]

2004 (1)

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 30–44(2004).
[CrossRef] [PubMed]

2003 (1)

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

2002 (1)

D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing(American Mathematical Society, 2002).

2001 (1)

G. Krüger and G. H. Glover, “Physiological noise in oxygenation-sensitive magnetic resonance imaging,” Magn. Reson. Med. 46, 631–677(2001).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221(1997).
[CrossRef]

1995 (1)

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transf. 53, 723–728(1995).
[CrossRef]

1994 (2)

M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246(1994).
[CrossRef]

A. Schwarz, “Three-dimensional reconstruction of temperature and velocity fields in a furnace,” Proceedings of ECAPT, The European Concerted Action on Process Tomography (International Society for Industrial Process Tomography, 1994), pp. 227–233.

1993 (2)

S. P. Juhlin, “Doppler tomography,” Proceedings of the 15th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, October 28–31, 1993, San Diego, California, USA (1993) pp. 212–213.
[CrossRef]

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33(1993).
[CrossRef] [PubMed]

1992 (3)

S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
[CrossRef] [PubMed]

V. A. Sharafutdinov, “Tomographic problem of photoelasticity,” Proc. SPIE 1843, 234–243(1992).
[CrossRef]

P. Juhlin, “Principles of Doppler Tomography,” Lund Institute of Technology, Sweden, Department of Mathematics, LUTFD2/(TFMA-92)/7002+17P, August (1992).

1991 (3)

H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833(1991).
[CrossRef]

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471(1991).
[CrossRef]

1990 (1)

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38(1990).
[CrossRef]

1988 (2)

S. J. Norton, “Tomographic reconstruction of 2-D vector fields: Application to flow imaging,” Geophys. J. Int. 97, 161–168 (1988).
[CrossRef]

G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212(1988).
[CrossRef] [PubMed]

1987 (2)

H. K. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924(1987).
[CrossRef] [PubMed]

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: Mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

1986 (2)

1983 (1)

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).

1982 (2)

W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Philos. Trans. R. Soc. London, Ser. A 307, 439–464 (1982).
[CrossRef]

S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201–233(1982).
[CrossRef] [PubMed]

1979 (1)

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” ISA Trans. 26, 2674–2677(1979).
[CrossRef]

1978 (2)

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 1978).
[CrossRef]

S. E. Segre, “The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave,” Plasma Phys. 20, 295–307(1978).
[CrossRef]

1977 (1)

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.

Aben, H.

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221(1997).
[CrossRef]

Aben, H. K.

Braun, H.

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471(1991).
[CrossRef]

Bronshtein, I. N.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Byer, R. L.

Cheney, W.

D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing(American Mathematical Society, 2002).

Ciuc, M.

Davis, A. M.

de Boor, C.

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 1978).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).

Efremov, N. P.

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transf. 53, 723–728(1995).
[CrossRef]

Ewart, T. E.

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833(1991).
[CrossRef]

Faghih, F.

Faris, G. W.

Flandro, G.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.

Giannakidis, A.

M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-d vector fields using a system of linear equations,” Proceedings of the 12th Annual Medical Image Understanding and Analysis Conference (MIUA 2008) July 2-3, 2008, Dundee, Scotland, UK, pp. 132–136.

Glover, G. H.

G. Krüger and G. H. Glover, “Physiological noise in oxygenation-sensitive magnetic resonance imaging,” Magn. Reson. Med. 46, 631–677(2001).
[CrossRef] [PubMed]

Greenleaf, J. F.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.

Habermann, C.

C. Habermann and F. Kindermann, “Multidimensional spline interpolation: Theory and applications,” Comput. Econ. 30, 153–169(2007).
[CrossRef]

Hauck, A.

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471(1991).
[CrossRef]

Hertz, H. M.

Howe, B. M.

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: Mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

Izatt, J. A.

Johnson, S. A.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.

Juhlin, P.

P. Juhlin, “Principles of Doppler Tomography,” Lund Institute of Technology, Sweden, Department of Mathematics, LUTFD2/(TFMA-92)/7002+17P, August (1992).

Juhlin, S. P.

S. P. Juhlin, “Doppler tomography,” Proceedings of the 15th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, October 28–31, 1993, San Diego, California, USA (1993) pp. 212–213.
[CrossRef]

Kadyrov, A.

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1631–1645(2006).
[CrossRef] [PubMed]

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 30–44(2004).
[CrossRef] [PubMed]

Kaiser, J.

T. D. Sanger, J. Kaiser, and B. Placek, “Reaching movements in childhood dystonia contain signal-dependent noise,” J. Child Neurol. 20, 489–496(2005).
[PubMed]

Kharchenko, V. N.

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transf. 53, 723–728(1995).
[CrossRef]

Kincaid, D.

D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing(American Mathematical Society, 2002).

Kindermann, F.

C. Habermann and F. Kindermann, “Multidimensional spline interpolation: Theory and applications,” Comput. Econ. 30, 153–169(2007).
[CrossRef]

Kramer, D. M.

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” ISA Trans. 26, 2674–2677(1979).
[CrossRef]

Kretzschmar, H.

H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).

Krüger, G.

G. Krüger and G. H. Glover, “Physiological noise in oxygenation-sensitive magnetic resonance imaging,” Magn. Reson. Med. 46, 631–677(2001).
[CrossRef] [PubMed]

Lauterbur, P. C.

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” ISA Trans. 26, 2674–2677(1979).
[CrossRef]

Linzer, M.

S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201–233(1982).
[CrossRef] [PubMed]

Muehlig, H.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Munk, W.

W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Philos. Trans. R. Soc. London, Ser. A 307, 439–464 (1982).
[CrossRef]

Musiol, G.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Norton, S. J.

S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
[CrossRef] [PubMed]

S. J. Norton, “Tomographic reconstruction of 2-D vector fields: Application to flow imaging,” Geophys. J. Int. 97, 161–168 (1988).
[CrossRef]

S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imaging 4, 201–233(1982).
[CrossRef] [PubMed]

Petrou, M.

M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-d vector fields using a system of linear equations,” Proceedings of the 12th Annual Medical Image Understanding and Analysis Conference (MIUA 2008) July 2-3, 2008, Dundee, Scotland, UK, pp. 132–136.

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1631–1645(2006).
[CrossRef] [PubMed]

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 30–44(2004).
[CrossRef] [PubMed]

Placek, B.

T. D. Sanger, J. Kaiser, and B. Placek, “Reaching movements in childhood dystonia contain signal-dependent noise,” J. Child Neurol. 20, 489–496(2005).
[PubMed]

Poluektov, N. P.

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transf. 53, 723–728(1995).
[CrossRef]

Puro, A.

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221(1997).
[CrossRef]

Rangayyan, R. M.

Rouseff, D.

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33(1993).
[CrossRef] [PubMed]

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833(1991).
[CrossRef]

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38(1990).
[CrossRef]

Sanger, T. D.

T. D. Sanger, J. Kaiser, and B. Placek, “Reaching movements in childhood dystonia contain signal-dependent noise,” J. Child Neurol. 20, 489–496(2005).
[PubMed]

Schwarz, A.

A. Schwarz, “Three-dimensional reconstruction of temperature and velocity fields in a furnace,” Proceedings of ECAPT, The European Concerted Action on Process Tomography (International Society for Industrial Process Tomography, 1994), pp. 227–233.

Schwarz, H. R.

H. R. Schwarz, Numerische Mathematik (B. G. Teubner, 1986).

Schwetlick, H.

H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).

Segre, S. E.

S. E. Segre, “The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave,” Plasma Phys. 20, 295–307(1978).
[CrossRef]

Semendyayev, K. A.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Sharafutdinov, V. A.

V. A. Sharafutdinov, “Tomographic problem of photoelasticity,” Proc. SPIE 1843, 234–243(1992).
[CrossRef]

Spindel, R. C.

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: Mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

Tanaka, M.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.

Tao, Y. K.

Winters, K. B.

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33(1993).
[CrossRef] [PubMed]

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833(1991).
[CrossRef]

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38(1990).
[CrossRef]

Worcester, P. F.

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: Mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

Wunsch, C.

W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Philos. Trans. R. Soc. London, Ser. A 307, 439–464 (1982).
[CrossRef]

Zahn, M.

M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246(1994).
[CrossRef]

Appl. Opt. (4)

Comput. Econ. (1)

C. Habermann and F. Kindermann, “Multidimensional spline interpolation: Theory and applications,” Comput. Econ. 30, 153–169(2007).
[CrossRef]

Geophys. J. Int. (1)

S. J. Norton, “Tomographic reconstruction of 2-D vector fields: Application to flow imaging,” Geophys. J. Int. 97, 161–168 (1988).
[CrossRef]

IEEE Trans. Dielectr. Electr. Insul. (1)

M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246(1994).
[CrossRef]

IEEE Trans. Image Process. (1)

S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 30–44(2004).
[CrossRef] [PubMed]

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1631–1645(2006).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471(1991).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33(1993).
[CrossRef] [PubMed]

Inverse Probl. (2)

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38(1990).
[CrossRef]

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221(1997).
[CrossRef]

ISA Trans. (2)

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3, pp. 3–15, 1977.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Tracing line A B unites two virtual sensors that reside at points A and B. The tracing line is defined by the two parameters ρ and θ (Radon domain coordinates) and goes through the digitized square reconstruction region of size 2 U × 2 U . The line segment is sampled with sampling step Δ s . The angle between the line segment and the positive direction of the x-axis is w. The size of the tiles with which we sample the 2-D space is P × P . Also shown is the unit vector s ̂ , which is parallel to line segment A B .

Fig. 2
Fig. 2

Comparison of the reconstruction performance for the cases when reconstruction was based on: (i) line-integral data from regularly placed sensors (RS) in relation to ( x , y ) coordinates; (ii) interpolated line-integral data obtained at virtual sensors that corresponded to uniform sampling of the Radon space and the employed interpolation method was the 1-D linear (IP1), the 1-D piecewise cubic spline (IP2), the piecewise cubic Hermite (IP3), the bilinear (IP4), the bicubic (IP5), and the 2-D spline (IP6); (iii) uniform sampling (US) of the parameter space using the actual measurements. The location of the source of the electric field was at (19, 19 ).

Fig. 3
Fig. 3

As in Fig. 2, but here the location of the source of the electric field was at ( 16 , 21).

Fig. 4
Fig. 4

As in Fig. 2, but here the location of the source of the electric field was at ( 21 , 12 ) .

Fig. 5
Fig. 5

As in Fig. 2, but here the location of the source of the electric field was at (24, 14.5).

Fig. 6
Fig. 6

Simulation results when the location of the source of the electric field was (from top to bottom) at (19, 19 ), ( 16 , 21), ( 21 , 12 ) , and (24, 14.5): (a) the recovered vector field when reconstruction was based on interpolated line-integral data (1-D piecewise cubic spline method) obtained at virtual sensors that corresponded to uniform sampling of the Radon space with Δ ρ = 0.5 and Δ θ = 1.5 ° ; (b) the theoretical electric field as computed from Coulomb’s law.

Fig. 7
Fig. 7

Comparison of the reconstruction performance in noisy environments for the cases: (i) when integral data from regularly placed sensors were used; (ii) when interpolated measurements that corresponded to uniform sampling with Δ ρ = 0.5 and Δ θ = 1.5 ° were used; (iii) when actual measurements that corresponded to uniform sampling with Δ ρ = 0.5 and Δ θ = 1.5 ° were used. (a), (b) Errors in vector field orientation and magnitude when noise was added to the measurements of 25% of the sensors, as a percentage of the true value. (c), (d) Errors in vector field orientation and magnitude when small perturbations in the sensor positions were added. Position perturbations were a percentage of the true positions. (e), (f) Errors in vector field orientation and magnitude when both sensors’ measurements and positions were changed by a percentage of their true values. In all cases, 25% of the sensors were perturbed.

Fig. 8
Fig. 8

As in Fig. 7, but 50% of the sensors were perturbed.

Fig. 9
Fig. 9

As in Fig. 7, but 75% of the sensors were perturbed.

Fig. 10
Fig. 10

As in Fig. 7, but all sensors were perturbed.

Fig. 11
Fig. 11

Comparison of the reconstruction performance in noisy environments for the cases: (i) when integral data from regularly placed sensors were used; (ii) when interpolated (1-D piecewise cubic spline) measurements that corresponded to uniform sampling with Δ ρ = 0.5 and Δ θ = 1.5 ° were used; and (iii) when actual measurements that corresponded to uniform sampling with Δ ρ = 0.5 and Δ θ = 1.5 ° were used. (a) Error in vector field orientation and (b) error in magnitude when Gaussian noise of zero mean was added to the measurements of the sensors.

Equations (14)

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ρ = x cos θ + y sin θ .
J 1 = A B f ¯ ( x , y ) s ̂ d s = A B f d s .
J i = l f ¯ l Δ s ¯ .
C g ¯ = b , ¯
C T C g ¯ = C T b ¯ ,
g ¯ = ( C T C ) 1 C T b ¯ .
S ( x ) = S ( x A ) + ( x x A ) ( S ( x B ) S ( x A ) ) ( x B x A ) .
S ( x ) = S i ( x ) = a i + b i ( x x i ) + c i ( x x i ) 2 + d i ( x x i ) 3 ,
S ( x ) = S i ( x ) = ( 1 + 2 t ) ( 1 t ) 2 S ( x i ) + t ( 1 t ) 2 h M ( x i ) + t 2 ( 3 2 t ) S ( x i + 1 ) + t 2 ( t 1 ) h M ( x i + 1 ) ,
S ( x , y ) = ( x 2 x ) ( y 2 y ) ( x 2 x 1 ) ( y 2 y 1 ) S Q 1 + ( x 2 x ) ( y y 1 ) ( x 2 x 1 ) ( y 2 y 1 ) S Q 2 + ( x x 1 ) ( y 2 y ) ( x 2 x 1 ) ( y 2 y 1 ) S Q 3 + ( x x 1 ) ( y y 1 ) ( x 2 x 1 ) ( y 2 y 1 ) S Q 4 .
S ( x , y ) = S i j ( x , y ) = k = 0 3 l = 0 3 a i j k l ( x x i ) k ( y y j ) l .
S ( x , y ) = S i j ( x , y ) = k = 1 ( N 1 ) + 3 l = 1 ( M 1 ) + 3 c i j k l u i k ( x ) v j l ( y ) .
u i k ( x ) = Φ ( x a h 1 + 2 i ) , v j l ( y ) = Φ ( y c h 2 + 2 j ) ,
Φ ( t ) = { ( 2 | t | ) 3 , if 1 < | t | < 2 4 6 | t | 2 + 3 | t | 3 , if 1 | 1 | 0 , elsewhere ,

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